ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu
ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO SECOND-ORDER DIFFERENTIAL EQUATIONS WITH
FRACTIONAL DERIVATIVE PERTURBATIONS
EVA BRESTOVANSK ´A, MILAN MEDVE ˇD
Abstract. In this article we study the asymptotic behavior of solutions to nonlinear second-order differential equations having perturbations that involve Caputo’s derivatives of several fractional orders. We find sufficient conditions for all solutions to be asymptotic to a straight line.
1. Introduction
The aim of this article is to study the asymptotic properties of solutions to scalar second-order ordinary differential equations that are perturbed with a term involving fractional derivatives. In these equations, the fractional derivatives most frequently used are the Riemann-Liouville and the Caputo’s fractional derivatives.
For basic definitions of fractional calculus and fundamentals of the theory of frac- tional differential equations, we refer the reader to the monographs [28, 29].
Fractional derivatives play the role of a damping force in vibrating systems in viscous fluids; which is the case in the well known Bargley-Torvik equation,
u00(t) +AcD3/2u(t) =au(t) +φ(t). (1.1) This equation models the motion of a rigid plate immersing in a viscous liquid with the fractional damping termAcD3/2u(t) which has Caputo’s fractional derivative (see [36]). Solutions of the linear fractionally damped oscillator equation with the Caputo’s derivative are analyzed in [26]. Existence results on boundary-value problems for the generalized Bagely-Torvik equation
u00(t) +AcDαu(t) =f(t, u(t),cDβu(t), u0(t)) (1.2) and for some other fractional differential equations can be found in [2, 3, 34, 27].
An existence and uniqueness result for the multi-fractional initial-value problem Au00+
N
X
k=1
BkcDαku(t) =f(t, u),
u(0) =u0, u0(0) =c1, 0< αk <2, k= 1,2, . . . , N
(1.3)
2000Mathematics Subject Classification. 34E10, 24A33.
Key words and phrases. Rimann-Liouville derivative; Caputo’s derivative;
fractional differential equation; asymptotic behavior.
c
2014 Texas State University - San Marcos.
Submitted June 16, 2014. Published September 26, 2014.
1
can be found in [33]. Caputo’s fractional derivatives in equation (1.3) play the role of damping terms. Abstract evolution equations with the Caputo’s fractional derivatives in the nonlinearities are studied in [13, 14] . Fractionally damped pen- dulums or oscillators are studied in [26, 33]. More articles devoted to this type of equations can be found in the list of references.
The following equation for a pendulum has the ordinary damping term λx0(t) and the fractional damping termsλ1cDβ1x(t), . . . , λmcDβmx(t):
x00(t) +λ1cDβ1x(t) +. . . λmcDβmx(t) +λx0(t) +ω2x(t) =g(t, x(t), x0(t)), wheret >0,βi∈(0,1),i= 1,2, . . . , m.
In [26], the equation
x00+λc0Dαx+ω2x= 0, x(0) =x0, x0(0) =x1, λ >0.
is analyzed by using the fractional version of the Laplace transformation. The Laplace image ofx(t) is
X(s) = sx0+x1+λsα−1x0
s2+λsα+ω2 ,
and the characteristic equation for the fractional differential equation is s2+λsα+ω2= 0.
Whenα=p/q this characteristic equation is was analyzed in [28]. For the linear fractionally damped oscillator withα= 1/2 the characteristic equation is
s2+λs1/2+ω2= 0,
whose analysis is much more complicated than in the case of the harmonic oscillator with the classical damping term (see [26]). It is clear that the exact analysis of linear fractional systems is extraordinary difficult. Some analysis and simulations of fractional-order systems can be found in the book [28]. The form of the equation (1.3) enables us to avoid some difficulties in the study of the stability problem by using a desingularization method developed in [19, 20, 22].
In the asymptotic theory of then-th order nonlinear ordinary differential equa- tions
y(n)=f(t, y, y0, . . . , y(n−1)), (1.4) a classical problem is to establish some conditions for the existence of a solution which approaches a polynomial of degree 1≤m≤n−1 ast→ ∞. The first paper concerning this problem was published by Caligo [7] in 1941. He proved that if
|A(t)|< k
t2+ρ (1.5)
for all large t, wherek, ρare given, then any solution y(t) of the linear differential equation
y00(t) +A(t)y(t) = 0, t >0, (1.6) can be represented asymptotically asy(t) = c1t+c2+o(1) when t →+∞, with c1, c2 ∈ R (see [1]). The first article on the nonlinear second-order differential equation
y00(t) +f(t, y(t)) = 0 (1.7)
was published by Trench [37] in 1963. Then there are publications by Cohen [9], Trench [37], Kusano and Trench [15] and [16], Dannan [12], Constantin [10] and [11], Rogovchenko [31], Rogovchenko [32], Mustafa, Rogovchenko [25], Lipovan [17]
and others. In the proofs of their results the key role is played by the Bihari in- equality [6] which is a generalization of the Gronwall inequality. Some results on the existence of solutions of then-th order differential equation approaching a polyno- mial function of the degreem with 1≤m≤n−1 are proved by Philos, Purnaras and Tsamatos [30]. Their proofs are based on an application of the Schauder Fixed Point Theorem. The paper by Agarwal, Djebali, Moussaoui and Mustafa [1] surveys the literature concerning the topic in asymptotic integration theory of ordinary differential equations. Several conditions under which all solutions of the one-dimensionalp-Laplacian equation
(|y0|p−1y0)0 =f(t, y, y0), p >1 (1.8) are asymptotic toa+btas t → ∞ for some real numbers a, bare proved in [24].
Some sufficient conditions for the existence of such solutions of the equation
(Φ(y(n))0 =f(t, y), n≥1, (1.9)
where Φ :R→Ris an increasing homeomorphism with a locally Lipschitz inverse satisfying Φ(0) = 0 are given in the paper [21].
In the papers [22, 23] the fractional differential equation of the Caputo’s type
cDαax(t) =f(t, x(t)), a≥1, α∈(1,2) (1.10) is studied. In [23] a higher order fractional differential equation is studied. In the both papers sufficient conditions under which all solutions of these equations are asymptotic to at+b, is proved. The problem of asymptotic integration of fractional differential equations of the Riemann-Liouville type is studied in [4, 5].
The obtained results are proved by an application of the fixed point method.
The aim of this paper is to give some conditions under which all solutions of a nonlinear second order differential equations perturbed by the Riemann-Liouville integral of a nonlinear function are asymptotic toat+b. The proof of this result is based on a desingularization method proposed by the author in the paper [19] (see also [20]).
2. second-order ODEs perturbed with a fractional derivative In this section we study the following fractional initial-value problem
u00(t) +f(t, u(t), u0(t)) +
m
X
i=1
ri(t) Z t
0
(t−s)αi−1hi(τ, u(τ), u0(τ))dτ = 0, (2.1)
u(1) =c1 u0(1) =c2, (2.2)
wheret >0 and 0< α <1.
Definition 2.1. A functionu: [0, T)→R, 0< T ≤ ∞,is called a solution of (2.1) ifu∈C2 on the interval (0, T), limτ→0+u(t) exists andu(t) satisfies (2.1) on the interval (0, T). This solution is called global if it exists for allt∈[0,∞).
We assume the following hypotheses:
(H1) Every solution of the equation (2.1) is global;
(H2) The functions f(t, u, v), hi(t, u, v), i = 1,2, . . . , m are continuous on D = {(t, u, v) :t∈[0,∞), u, v∈R}and the functionsri(t), i= 1,2, . . . , m are continuous on the interval [0,∞);
(H3) There exist continuous, nonnegative functions hi : [0,∞) →R, i= 1,2,3 and continuous, positive and nondecreasing functionsgj : [0,∞)→Rsuch that
|f(t, u, v)| ≤Se−γt h1(t)g1
|u|
t
+h2(t)g2(|v|) +h3(t)
, t >0, whereS, γ >0;
(H4) There exist continuous, nonnegative functions hij : [0,∞) → R, i = 1,2, . . . , m; j = 1,2,3 and continuous positive, nondecreasing functions Gij: [0,∞)→R,i= 1,2, . . . , m;j= 1,2,3 such that
|fi(t, u, v)| ≤h1i(t)Gij |u|
t
+h2i(t)G2i(|v|) +h3i(t), t >0;
for all (t, u, v)∈D,i= 1,2, . . . , m;
(H5) |ri(t)| ≤Sie−ωit,t≥0, whereSi>0,ωi>1,i= 1,2, . . . , m;
(H6) There exist numbers pi > 1,i = 1,2, . . . , m such thatpi(αi−1) + 1> 0 with
Z ∞
0
hi(s)q <∞, Z ∞
0
hij(s)q <∞, i= 1,2, . . . , m; j= 1,2,3, whereq=q1q2. . . qm,qi=pi/(pi−1),i= 1,2, . . . , m;
(H7)
Z ∞
0
τq−1dτ ω(τ) =∞, where
ω(w) =g1(w)q+g2(w)q+
m
X
i=1 2
X
j=1
Gij(w)q.
Theorem 2.2. If the conditions (H1)–(H7) are satisfied then for every global so- lution u(t) of (2.1) there exist real numbers a, b such that u(t) =at+b+o(t) as t→ ∞.
For the proof of this theorem we use the following lemma, proved in [19].
Lemma 2.3. Let pj, αj,j= 1,2, . . . , msatisfy (H4). Then Z t
0
(t−s)pj(αj−1)epjsds≤Qjepjt, t≥0, j= 1,2, . . . , m, where
Qj =Γ(1 +pj(αj−1)) p1+pj(αj−1) , and
Γ(x) = Z ∞
0
sx−1e−sds, x >0 which is the Euler gamma function.
Proof of Theorem 2.2. Letu(t) be a solution of (2.1) corresponding to the initial conditions (2.2). Then
u0(t) =c2− Z t
1
f(s, u(s), u0(s))ds
−
m
X
i=1
Z t
1
ri(s) Z s
0
(s−τ)αi−1fi(τ, u(τ), u0(τ))dτ ds,
(2.3)
u(t) =c1+c2(t−1)− Z t
1
(t−s)f(s, u(s), u0(s))ds
−
m
X
i=1
Z t
1
(t−s)ri(s)Z s 0
(s−τ)αi−1fi(τ, u(τ), u0(τ))dτ ds.
(2.4)
From conditions (H3)–(H5) it follows that fort≥1,
|u0(t)| ≤ |c2|+ Z t
1
[h1(s)g1
|u(s)|
s
+h2(s)g2(|u0(s)|) +h3(s)]ds
+
m
X
i=1
Z t
1
|ri(s)|
Z s
0
(s−τ)αi−1h
h1i(τ)G1i
|u(τ)|
τ
+h2i(τ)G2i(|u0(τ)|) +h3i(τ)i dτ ds and
|u(t)|
t ≤C+ Z t
1
[h1(s)g1
|u(s)|
s
+h2(s)g2(|u0(s)|) +h3(s)]ds
+
m
X
i=1
Z t
1
|ri(s)|
Z s
0
(s−τ)αi−1h
h1i(τ)G1i
|u(τ)|
τ
+h2i(τ)G2i(|u0(τ)|) +h3i(τ)i dτ ds,
where C =|c1|+|c2|. Ifqi =pi/(pi−1) then using Lemma 2.3 and the H¨older inequality we estimate
Z s
0
(s−τ)αi−1k1i(τ)G1i
|u(τ)|
τ dτ
≤Z s 0
(s−τ)pi(αi−1)epiτdτ1/piZ s 0
e−qiτh1i(τ)qiG1i
|u(τ)|
τ qi
dτ1/qi
≤QiesZ s 0
e−qiτh1i(τ)qiG1i
|u(τ)|
τ qi
dτ1/qi
, Z s
0
(s−τ)αi−1h2i(τ)G2i(|u0(τ)|)dτ ≤QiesZ s 0
e−qiτh2i(τ)qiG2i(|u0(τ)|)qidτ1/qi
, Z s
0
(s−τ)αi−1h3i(τ)dτ ≤QiesZ s 0
e−qiτh3i(τ)qidτ1/qi
. These inequalities yield
|u(t)|
t ≤C+S Z t
1
e−γs h1(s)g1
|u(s)|
s
+h2(s)g2(|u0(s)|) +h3(s) ds +
m
X
i=1
SiQi
Z t
1
e−(ωi−1)snZ s 0
e−qiτh1i(τ)qiG1i
|u(τ)|
τ qi
dτ1/qi
+Z s 0
e−qiτh2i(τ)qiG2i(|u0(τ)|)qidτ1/qi
+Z s 0
e−qiτh3i(τ)qidτ1/qio ds Sinceωi>1 andγ >0, we have the estimate
|u(t)|
t ≤C+S Z t
0
e−γs h1(s)g1
|u(s)|
s ) +h2(s)g2(|u0(s)|) +h3(s) ds
+
m
X
i=1
Si Qi ωi−1
nZ t
0
e−qiτh1i(τ)qiG1i |u(τ)|
τ qi
dτ1/qi
+Z t 0
e−qiτh2i(τ)qiG2i(|u0(τ)|)qidτ1/qi
+Z t 0
e−qiτh3i(τ)qidτ1/qi
dτo . Denoting byz(t) the right-hand side of this inequality, we have
|u(t)|
t ≤z(t), |u0(t)| ≤z(t), t≥0.
Sinceg1, g2, G1i, G2i, G3i are nondecreasing functions these inequalities yield z(t)≤C+S
Z t
0
e−γs
h1(s)g1(z(s)) +h2(s)g2(z(s)) +h3(s) ds
+
m
X
i=1
Si
Qi
ωi−1 nZ t
0
e−qiτh1i(τ)qiG1i(z(τ))qidτ1/qi
+Z t 0
e−qiτh2i(τ)qiG2i(z(τ))qidτ1/qi
+Z t 0
e−qiτh3i(τ)qidτ1/qi
dτo . Let Q = max{SωiQi
i−1, i = 1,2, . . . , m} and q = q1q2. . . qm. Then using the in- equality (P3m+2
i=1 ai)q ≤(3m+ 2)q−1(P3m+2
i=1 aqi) for any nonnegative numbersai, i= 1,2, . . . ,3m+ 2, we obtain the estimate
z(t)q
≤(3m+ 2)q−1
Cq+Sq Z t
1
e−γsZ t 1
(h1(s)g1(z(s)) +h2(s)g2(z(s)) +h3(s))dsq +Qq
m
X
i=1
nZ t
0
e−qiτh1i(τ)qiG1i(z(τ))qidτqˆi
+Z t 0
e−qiτh2i(τ)qiG2i(z(τ))qidτqˆi
+Z t 0
e−qiτh3i(τ)qidτqˆi
dτo , where ˆqi=q1q2. . . qi−1qi+1. . . qm. If ˆpi= ˆqqˆi
i−1 andp= q−1q , then using the H¨older inequality we obtain the following inequalities
Z t
0
e−γsnZ s 1
h1(τ)g1(z(τ)) +h2(τ)g2(z(τ)) +h3(τ) dτoq
ds
≤ 1 pγ
1/pZ t 0
h1(s)g1(z(s)) +h2(s)g2(z(s)) +h3(s)q ds
≤3q−1 1 pγ
1/p Z t
0
h1(s)qg1(z(s))q+h2(s)qg2(z(s))q+h3(s)q ds, Z t
0
e−qiτh1i(τ)qiG1i(z(τ))qidτqˆi
≤Z t 0
e−pˆiqisdspiˆ1Z t 0
h1i(s)qG1i(z(s))qds
≤ 1
(ˆpiqi−1)1/pˆi Z t
0
h1i(s)qG1i(z(s))qds,
Z t
0
e−qiτh2i(τ)qiG2i(z(τ))qidτqˆi
≤ 1
(ˆpiqi−1)1/pˆi Z t
0
h2i(s)qG2i(z(s))qds, Z t
0
e−qish3i(s)qids≤ 1 (ˆpiqi−1)1/ˆpi
Z t
0
h3i(s)qds.
From these inequalities and (H6) it follows that there exist a constantA >0 such that
z(t)q ≤A+A Z t
0
[h1(s)qg1(z(s))q+h2(s)qg2(z(s)) +h3(s)q]ds +A
m
X
i=1
Z t
0
h1i(s)qG1i(z(s))qds+A
m
X
i=1
Z t
0
h2i(s)qG2i(z(s))qds.
This inequality implies that the functionv(t) =z(t)q satisfy the inequality
v(t)≤A+ Z t
0
F(s)ω(v(s)1q)ds, t≥0, where
ω(z) =g1(z)q+g2(z)q+
m
X
i=1
[G1i(z)q+G2i(z)q],
F(t) =A
h1(t)q+h2(t)q+
m
X
i=1
[h1i(t)q+h2i(t)q] .
From (H6) it follows thatR∞
0 F(s)ds <∞, and from the Bihari inequality we obtain v(t)≤K0= Ω−1[Ω(A) +
Z ∞
0
F(s)ds]<∞, t≥0, where
Ω(u) = Z v
v0
σ ω(σ). Note that Ω(A) +R∞
0 F(s)dsis always in the range of Ω−1, asω(∞) =∞by (H7).
This implies that there is a constantK >0 such that
|u0(t)| ≤z(t)≤K, |u(t)|
t ≤z(t)≤K, t≥0.
In conclusion, we obtain the existence of the limit
t→∞lim
|u(t)|
t =c,
which completes the proof.
3. Example
The following example is a fractional modification of the Caligo’s example men- tioned in the introduction.
u00(t) +Se−γtn
ω2 1
(t+ 1)1+1q u(t)
t
+k1
1 (t+ 1)1+1q
u0(t) +k2
1 t1+1q
o
+
m
X
i=1
Sie−ωit Z t
0
(t−s)αi−1n η1i
(s+ 1)1+qi1
ln u(s) s
qi
+ 21/qi
+ η2i (s+ 1)1+qi1
ln
u0(s)]qi+ 21/qi
+ η3i (s+ 1)1+qi1
o ds= 0,
(3.1)
whereS,γ,ω,k1,k2,η1i,η2i, η3i, i= 1,2, . . . , m are positive numbers and γ, ωi, q,qi,αi satisfy the conditions in Theorem 2.2. Here
hi(t) = ki (t+ 1)1+1q
, hji(t) = ηji (t+ 1)1+qi1
,
i = 1,2, . . . , m, j = 1,2,3, g1(u) = g1(u) = [ln(uq + 2)]1q, g1i(u) = g2i(u) = [ln(uqi+ 2)]1/qi. Since
Z ∞
0
hi(s)qds= Z ∞
0
1
(s+ 1)1+qds=1 q and
Z ∞
0
σq−1dσ g1(σ)q =
Z ∞
0
σq−1dσ [ln(σq+ 2)] = 1
q Z ∞
0
dτ
ln(τ+ 2) =∞,
all conditions of Theorem 2.2 are satisfied and therefore for any solution of (3.1) there exist constantsa, b∈Rsuch thatu(t) =at+b+o(t) ast→ ∞.
Acknowledgements. This research was supported by the Slovak Grant Agency VEGA-MˇS, project No. 1/0071/14.
References
[1] R. P. Agarwal, S. Djebali, T. Moussaoui, O. G. Mustafa;On the asymptotic integration of nonlinear differential equations, Journal of Comput. and Applied Math. , 202 (2007), 352-376.
[2] R. P. Agarwal, D. O’Regan, S. Stanˇek;Positive solutions for Dirichlet problems of singular fractional differential equations, J. Math. Anal. Appl. 371 (2010), 57-68.
[3] R. P. Agarwal, D. O’Regan, S. Stanˇek; Positive solutions for mixed problems of singular fractional differential equations, Mathematische Nachrichten, 11.
[4] D. Bˇaleanu, O. G. Mustafa, R. P. Agarwal;Asymptotically linear solutions for some linear fractional differential equations, Abstr. Appl. Anal., Vol. 2010, Article ID 865139, 8 p.
[5] D. Bˇaleanu, O. G. Mustafa R. P. Agarwal;Asymptotic integration of(1 +α)−order fractional differential equations, Comput. Math. Appl. 62 (2011), 1492-1500.
[6] I. Bihari; Researches of the boundedness and stability of solutions of non-linear differential equations, Acta Math.Acad.Sci.Hung., Vol. 8 (1957), 261-278(2011), 1-15.
[7] D. Caligo; Comportamento asintotico degli integrali dell’equazione y00(x) +A(x)y(x) = 0 nell’ipotesilimx→+∞A(x) = 0, Boll.Un.Mat.Ital., Vol. 6(1941),286-295.
[8] M. Caputo;Linear models of dissipation whoseQis almost frequency independent II, Geo- phys. J. Royal Astronom. Soc., Vol. 13 (1967), 529-535.
[9] D. S. Cohen; The asymptotic behavior of a class of nonlinear differntial equations, Proc.
Amer. Math. Soc. Vol 18, (1967),607–609.
[10] A. Constantin;On the asymptotic behavior of second order nonlinear differential equations, Rend. Math. Appl., Vol. 13, No. 7 (1993), 627–634.
[11] A. Constantin;On the existence of positive solutions of second order differential equations, Annali di Matematica, Vol. 184 (2005), 131-138.
[12] F. M. Dannan;Integral inequalities of Gronwall-Bellman-Bihari type and asymptotic behavior of certain second order nonlinear differential equations, J. Math. Anal. Appl., Vol. 108 (1985), 151–164.
[13] M. Kirane, M. Medveˇd, N. E. Tatar; Semilinear Volterra integrodifferential problems with fractional derivatives, Abstract and Applied Analysis, 2011 (2011), Article ID 510314.
[14] M. Kirane, M. Medveˇd, N. E. Tatar;On the nonexistence of blowing-up solutions to a frac- tional functional-differential equations, Georgia J. Math. 19 (2012), 127-144.
[15] T. Kusano, W. F. Trench;Global existence of second order differential equations with inte- grable coefficients, J. London Math. Soc., Vol. 31 (1985), 478–486.
[16] T. Kusano, W. F. Trench;Existence of global solutions with prescribed asymptotic behavior for nonlinear ordinary differential equations, Mat. Pura Appl. Vol. 142(1985), 381-392.
[17] O. Lipovan;On the asymptotic behaviour of the solutions to a class of second order nonlinear differential equations, Glasg. Math. J. 45 no. 1, (2003), 179–187.
[18] Q.-H. Ma, J. Peˇcariˇc, J.-M. Zhang; Integral inequalities of systems and the estimate for solutions of certain nonlinear two-dimensional fractional differential systems, Computers and Mathematics with Applications, Vol. 61 (2011), 3258-3267.
[19] M. Medveˇd; A new approach to an analysis of Henry type integral inequalities and their Bihari type versions, J. Math. Anal. Appl. Vol. 214(1997)349-366.
[20] M. Medveˇd;Integral inequalities and global solutions of semilinear evolution equations, J.
Math. Anal. Appl., Vol. 37, No. 4 (2002), 871–882.
[21] M. Medveˇd, T. Moussaoui; Asymptotic integration of nonlinear Φ−Laplacian differential equations, Nonlinear Analysis, TMA, Vol. 72 (2010), 1-8.
[22] M. Medveˇd;On the asymptotic behavior of solutions of nonlinear differential equations of integer and also of non-integer order, EJQTDE, Proc. 9th Coll. QTDE, No. 10, 1-9.
[23] M. Medveˇd;Asymptotic integration of some classes of fractional differential equations, Tatra Mountains Math. Publ., Vol. 54 (2013), 1-14.
[24] M. Medveˇd, E. Pek´arkov´a; Long time behavior of second order differential equations with p-Laplacian, Electronic J. Diff. Eqs., Vol. 2008 (2008), 1-12.
[25] O. G. Mustafa, Y. V. Rogovchenko;Globa differential equations existence of solutions with prescribed asymptotic behavior for second-order nonlinear differential equations, Nonl. Anal.
TMA ,Vol.51 (2002), 339-368.
[26] M. Naber; Linear fractionally damped oscillator, International J. of Diff. Eqs. 2010 (2010), ID 197020.Linear fracti
[27] D. O’Regan, S. Stanˇek;Fractional boundary value problems with singularities in space vari- ables, Nonlinear Dyn., 71 (2013), 641-652.
[28] I. Petr´aˇs; Fractional-order Nonliner System Modeling, Analysis and Simulation, Higher Education Press, Springer 2011.
[29] I. Podlubny;Fractional Differential Equations, Academic Press, San Diego 1999.
[30] Ch. G. Philos, I. K. Purnaras, P. Ch. Tsamatos;Large time asymptotic to polynomials solu- tions for nonlinear differential equations, Nonl. Anal. TMA , Vol. 59 (2004), 1157–1179 [31] Y. V. Rogovchenko;On asymptotics behavior of solutions for a class of second order nonlin-
ear differential equations, Collect. Math., Vol. 49, No.1 (1998), 113-120.
[32] S. P. Rogovchenko, Yu. V. Rogovchenko;Asymptotics of solutions for a class of second order nonlinear differential equations, Portugaliae Math., Vol. 57, No.1 (2000), 17-32.
[33] M. Seredynska, A. Hanyga; Nonlinear differential equations with fractional damping with applications to 1dof and 2dof pendulum, Acta Mechanica, 176 (2005), 169-183.
[34] S. Stanˇek;Two-point boundary value problems for the generalized Bagley-Torvik fractional differential equations, Centr. Eur. J. Math. 11(3) (2013), 574-592.
[35] J. Tong; The asymptotic behavior of a class of nonlinear differential equations of second order, Proc. Amer. Math. Soc., Vol. 84 (1982), 235-236.
[36] P. J. Torvik, R. L. Bagley;On the appearance of the fractional derivatives in the bhavior of real materials, Trans. ASME J. Appl. Mech. 51 (2) (1984), 294-298.
[37] W. F. Trench; On the asymptotic behavior of solutions of second order linear differential equations, Proc. Amer. Math. Soc.,Vol. 54 (1963), 12-14.
Eva Brestovansk´a
Department of Economics and Finance, Faculty of Management, Comenius University, Odboj´arov str., 831 04 Bratislava, Slovakia
E-mail address:[email protected]
Milan Medveˇd
Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathe- matics, Physics and Informatics, Comenius University, 842 48 Bratislava, Slovakia
E-mail address:[email protected]
